Auction Design without Commitment

Feasible choice rules To highlight the fact that in the above definition the
English auction is conditioned on the prior p, denote it by φ^E [p]. Now the function
φ^E : ∆(Θ) → Φ can be taken as the the English auction -choice rule. Construct a
correspondence C^φE ⊂ VIC such that

C^φ [p] = ©Ф ∈ VIC[p] : Ф is not ex post dominated by φE[p⁰], for any p' ∈ ∆(Θ)} .

Note that the English auction -choice rule is idempotent: φ^E [p(x, φ^E [p])] =
1_x , for all x ∈ φ^E [p](supp(p)), for all p. That is, after running the English auction,
a new English auction does not change the outcome. This implies that the English
auction φ^E [p] is not ex post dominated by φ^E[p(x, φ^E [p])]. Hence it follows that
φ^E [p] ∈ C^φE [p] for all p. More compactly:

Lemma 3 φE [∙] is consistent.

If φ^E [p] also maximizes v in C^φE [p] for all p, then φ^E satisfies the one-deviation
property. We now show that this necessarily holds: any mechanism in C^φE [p] is
outcome equivalent - and hence payoff equivalent - with the English auction φ^E [p].

Lemma 4 φ ∈ C^φE [p] only if φ is outcome equivalent to φ^E [p], for all p.

Proof. Relegated to Appendix A. ■

That is, given p, the only veto-incentive compatible mechanisms that are not ex
post dominated by any φ^E [p⁰] are the mechanism φ^E [p] itself and its versions that
may additionally reveal some non-relevant information concerning the winner’s
type. Thus φ^E has a "fixed point" property.

We next demonstrate that any stationary σ that is consistent and meets
the one-deviation property induces C^σ that contains φ^E , specified for some tie-
breaking rule w. Heuristically, if φ ex post dominates φ^E [p], then φ must change
φ^E [p]’s allocation. Since any outcome x of φ^E [p] reveals the winner and his least
possible valuation given the other buyers’ valuations, φ must threaten the win-
ner to sell the good to the buyer with the second highest valuation to force the
winner to pay a higher price. However, the threat is not credible since when the
winner declines the offer, the seller sells, by stationarity, to the winner with his
least possible valuation.

Lemma 5 Let a stationary choice rule σ be consistent and satisfy the one-deviation
property. Then there is a tie-breaking rule w such that φ^E [p] ∈ C^σ [p], for all p.

Proof. Construct w as follows: For any θ ∈ Θ, denote by 1θ the degenerate
prior such that supp(1θ) = {θ}. Then there is an outcome xθ such that σ[1θ] = 1x_θ .
Let w(θ)=i if xθ allocates the good to i. By Lemma 2, such w(θ) satisfies (5).
Use this w to construct φ^E .

Suppose, to the contrary of the claim, that there is p such that φ^E [p] 6∈ C^σ [p].
Then φ^E[p] is ex post dominated by σ[p(x, φ^E [p])] for some x ∈ φ^E[p](supp(p)).
Denote σ[p(x, φ^E [p])] = g ◦ h.

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